This method is based on matching the boundary fields with the constitutive properties of the wall material. Et Transverse electric fields in lossy waveguides Ets Transverse electric fields in lossless waveguides Zs Surface impedance of the wall material.
CHAPTER 1
Scientific Motivation
For example, Stratton's equation (Stratton, 1941) can only be applied to circular waveguides, but not to rectangular waves. This is an added advantage as the formulation can be applied at higher frequencies (millimeter and submillimeter wavelengths).
Technological Background
Layout of the quartz substrate with the SIS mixer built on it (Vassilev and Belitsky, 2001b). The RF and LO signals are then fed to each of the mixer terminal tuning circuits.
Overview of Thesis
To calculate the finite thickness of the strip and the ground plane, the surface impedance formulated by Kerr (1999) is included in the equation. To account for the dissipative effect of a lossy CPW, the frequency-dependent effective dielectric constant εeff given by Hasnain et al. 1986) is included in the loss equation.
CHAPTER 2
Introduction
Papadopoulos' perturbation method (PPM) shows that the attenuation at frequencies well above fc remains in close agreement with that calculated using the power loss method for non-degenerate modes. In addition, at very high frequencies – especially those approaching millimeter and submillimetre wavelengths – the loss tangent of the conductive wall decreases.
General Wave Behaviours along Uniform Guiding Structures
A TM wave consists of a non-zero electric field, but a zero magnetic field in the longitudinal direction. iii) Transverse electric waves (TE). A TE wave consists of a zero electric field but a nonzero magnetic field in the longitudinal direction.
Fields in Cartesian Coordinates
The transverse field components can be derived by substituting the longitudinal field components into Maxwell's equations without convolution of the source. If the transverse components of the field are expressed in terms of the longitudinal components of the field Ez and Hz, the following equations can be obtained (Cheng, 1989).
A Review of Some Conventional Methods
- The Power-Loss Method
- Papadopoulos’ Perturbation Method
A characteristic equation can then be derived by subtracting both equations and subsequently integrating the result over the cross section of the waveguide. For the root, where Amn > Amn', the propagation constant kz corresponds to a perturbed TEmn state.
The Proposed Method
- Fields in a Lossy Rectangular Waveguide
- Constitutive Relations for TE and TM Modes
To obtain non-trivial solutions to (2.46) and (2.47), the determinant of the equations must be zero. By canceling the determinant of the coefficients of E0 and H0 in (2.46) and (2.47), the following transcendental equations are obtained.
HFSS Simulation
Once the setup is complete as shown in Figure 2.4, the attenuation constant αz of the dominant TE10 mode is then obtained by simulating the model at a range of frequencies. As shown in Figure 2.5, the loss predicted by the proposed method is closely consistent with the simulated loss result.
Experimental Setup
Thus, a short circuit at the end of this groove turns into an open circuit at the contact point between the choke and the flanges. Since the voltage drop at the ohmic contact between the flanges and the choke is negligible, voltage breakdown is avoided (Pozar, 2005).
Results and Discussion
As shown in Figure 2.16, for frequencies beyond millimeter wavelengths, the loss calculated by the proposed method appears to be higher than those by the power loss and PPM methods. At such high frequencies, the wave propagating in the waveguide is a hybrid mode and the presence of the longitudinal electric field Ez can no longer be neglected. Here, the power loss method can only give αz, while PPM and the proposed method give both βz and αz.
In Figures 2.18 and 2.19, αz calculated by the PPM and the proposed method agree very well near the limit value.
PPM) (The proposed
The attenuation αz of the degenerate modes TE11 and TM11 is shown in Figures 2.18 to 2.21, both near the limit value and in the propagation region. However, Figures 2.20 and 2.21 show that as the frequency increases above 28.5 GHz for the TE11 and 27.0 GHz for the TM11, the results begin to diverge significantly.
PPM)
Summary
The formulation is based on the matching of the tangential electric and magnetic fields at the boundary of the wall, and allows the wavenumbers to take complex values. The electromagnetic fields are used in conjunction with the concept of surface impedance to derive transcendental equations, the roots of which give values for the wavenumbers in the x and y directions for different TE or TM modes. The calculated attenuation curves agree well with the PPM and experimental results for the case of the dominant TE10 mode.
In other words, the total loss of two coexisting modes is greater than the sum of the losses of two modes propagating independently.
CHAPTER 3
- Introduction
- Fields in Circular Cylindrical Coordinates
- A Review of Stratton’s Approach
- The Proposed Method
- HFSS Simulation
- Experimental Setup
- Results and Discussion
- Summary
From (2.38), the surface impedance Zs at the wall boundary (r = ar) can be expressed as Solving the determinants of the coefficients Cn and Cn' in (3.16) results in the following transcendental equation. A further simulation experiment shows that the loss and the cut-off frequency decrease as the radius ar of the waveguide increases.
The S21 parameter of the dominant TE11 mode was measured from the VNA. a) Hollow circular waveguides made of brass, (b) a taper, (c) a circular choke and (d) a circular to rectangular waveguide junction.
CHAPTER 4
- Introduction
- Properties of Superconductors
- The Semiconductor Picture of the Superconductor
- The Complex Conductivity
- Characteristic Equations for Superconducting Waveguides
- Results and Discussion
- Summary
The attenuation of the superconducting waveguides above fg eventually exceeds the attenuation of the waveguide operating at room temperature. For frequencies below the gap frequency fg, the skin depth of the superconducting Nb is of the order of 10−8 m, which is much smaller than in the normal state. The surface resistance Rs of the waveguide can be calculated by substituting the values of the complex conductivity into the real part of Zs in (2.38).
The attenuation constants are calculated by substituting the values of the complex conductivity into the transcendental equations formulated in Chapters 2 and 3.
CHAPTER 5
Introduction
Full-wave analysis allows the coexistence of longitudinal fields, thus taking into account the dispersive nature of the microstrip lines. In this chapter, a new full-wave analysis approach is presented that takes into account the dispersive nature of the microstrip structures. Thus, for convenience, some of the available quasi-static methods will be briefly discussed, before discussing the new method of full-wave analysis.
In the following sections, the new method will also be demonstrated to give more realistic values especially for superconducting microstrip lines operating in the millimeter and submillimeter regimes, where the wavelengths are comparable to the dimensions of the microstrip structures.
Methods to Compute Microstrip Loss
- Formulations based on the Incremental Inductance Rule
- Formulations based on the Transmission Line Model
The quantity ∆w can be found by comparing the result of the conformal map of a microstrip line with strip thickness ts ≠ 0 and ts. According to Wheeler (1964), a homogeneous medium can be introduced to replace air and the dielectric substrate of the dielectric-filled microstrip line. The dielectric constant of the medium is represented as εeff and is known as the "effective dielectric constant".
By including χ in the Matic equation and replacing the dielectric constant εr of the substrate with the effective dielectric constant εeff, the attenuation constant α and the phase constant β are read.
The Proposed Method
- Fields in the Dielectric Substrate
Inserting the center strip into the waveguide causes currents to flow in the x and z directions on the strip. Thus, the mathematical problem can be simplified by dealing with only half of the microstrip structure. Nevertheless, due to the finite conductivity of the strip and base material, both Et and .
To account for the different fields at the boundary of the strip and the ground plane with different surface impedance, a phase parameter φy is introduced.
Fields in Free Space
As shown in Figure 5.2, the air or free space region is unbounded in the y direction. In the x-direction, the fields must satisfy the boundary condition of perfectly conducting walls in x. By solving the homogeneous Helmholtz equation and applying the boundary condition, the longitudinal fields can be expanded in this way.
Using the same procedure used to derive the transverse fields in the dielectric substrate, we obtain the following expressions for the transverse fields in the free space region.
Characteristic Equation for Microstrip Lines
It is worth noting that the right-hand side of (5.24) is multiplied by two because only the right-hand half of the microstrip structure is considered to calculate the total sum-of-fields ratio. If we let the determinants of the coefficients Ed and Hd in (5.25) cancel, we obtain the following group of transcendental equations. The propagation constant kz for each mode can be expressed in terms of transverse wavenumbers using one of the dispersion relations shown in (5.27) and (5.28) below.
As in the case of lossy waves, the Powell Hybrid root search algorithm in a NAG routine is used to find the root of kyd.
The Superconducting Microstrip Lines
Results and Discussion
Loss in a Nb microstrip line at room temperature and f = 100 GHz as a function of the ratio of strip thickness to substrate height (ts/b). A comparison of the attenuation constant and phase velocity of the Nb microstrip line at room temperature and at T = 4.2 K is shown in Figures 5.12 and 5.13. As can be clearly seen from Figure 5.12, at f below fg the attenuation of the superconducting microstrip line is much smaller than at room temperature.
Above fg, both the attenuation and the phase velocity approach those obtained from the microstrip line at room temperature.
Summary
CHAPTER 6
- Introduction
- Attenuation in Coplanar Waveguides
- Comparison between Microstrip Lines and Coplanar Waveguides
Some of the steps that can be taken to minimize energy leakage are listed below: From Figure 6.2, it can be clearly seen that as q increases, the conduction loss of the microstrip line decreases at a higher rate than CPW. At large dimensions where q > 2.2, the microstrip line loss is much lower than CPW.
However, at q < 2.2 it can be observed that the conduction loss of the CPW appears to be significantly lower.
Microstrip
Summary
A comparison between the attenuation of waves propagating in a microstrip line and coplanar waveguide (CPW) is performed. The results for the normal and superconducting cases show that the conduction loss of a microstrip line decreases at a higher rate than the CPW with increasing dimensions for both devices. As the dimensions are reduced to that comparable to the wavelength (i.e. q < 2.2), the loss in a CPW appears to be significantly lower.
The preliminary result illustrated in Figures 6.2 and 6.4 actually suggests that CPWs can be considered as a better alternative device in millimeter and submillimeter wave coupling.
CHAPTER 7
- Future Work
- Full-Wave Analysis of Coplanar Waveguides
- Bending Losses in Rectangular Waveguides
- Input Impedance of a Microstrip Probe in Circular Waveguides
The additional loss obtained with the new method is induced by the mode coupling effect of the simultaneous propagation of modes. Since Matick assumes that the strip is infinitely wide, the additional loss found in the new method can therefore be attributed to the edge loss at the edges of the strip. Proceedings of the 5th International Symposium on Space Terahertz Technology, University of Michigan, Ann Arbor, Michigan.
Proceedings of the 12th International Symposium on Space Terahertz Technology, Shelter Island, San Diego, California.
